A diagram approach to the strong coupling in the single-impurity Anderson model

We propose a diagram theory around the atomic limit for the single-impurity Anderson model in which the strongly correlated impurity electrons hybridize with free (uncorrelated) conduction electrons. Using this diagram approach, we prove a linked-cluster theorem for the vacuum diagrams and derive Dyson-type equations for localized and conduction electrons and the corresponding equations for mixed propagators. The system of equations can be closed by summing an infinite series of ladder diagrams containing irreducible Green’s functions. The result allows discussing resonances associated with quantum transitions at the impurity site. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 155, No. 3, pp. 474–497, June, 2008.     

Perturbation theory for the periodic Anderson model

We investigate the system of conductivity electrons and f-localized electrons described by the periodic Anderson model. Single-site hybridization of the state of two constituent subsystems of electrons is treated as a perturbation. We develop a new diagram technique based on the use of multiparticle one-site irreducible Green’s functions for the f-electrons and the standard Wick theorem for the subsystem of conductivity electrons. We derive the Dyson equations for the one-particle Green’s functions and find the relation between these functions. These results are exact and can be used as a starting point for self-consistent approximations. In the Hubbard-I approximation, we analyze the spectrum of one-particle perturbations. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 110, No. 2, pp. 308–322, February, 1997.     

Diagram theory for the periodic anderson model: Stationarity of the thermodynamic potential

We develop a diagram theory for the periodic Anderson model assuming that the Coulomb repulsion of localized f electrons is the main parameter of the theory. The f electrons are strongly correlated and the c conduction electrons are uncorrelated. We determine the f-electron correlation function and the c-electron mass operator. We formulate the Dyson equation for c electrons and a Dyson-type equation for f electrons and their propagators. We define the skeleton diagrams for the correlation function and the thermodynamic functional. We establish the stationarity of the renormalized thermodynamic potential under variation of the mass operator. The obtained results are applicable to both the normal and the superconducting system states.     

Two-time Green’s functions in superconductivity theory

Based on the method of the equations of motion for two-time Green’s functions, we derive superconductivity equations for different types of interactions related to the scattering of electrons on phonons and spin fluctuations or caused by strong Coulomb correlations in the Hubbard model. We derive an exact Dyson equation for the matrix Green’s function with the self-energy operator in the form of the multiparticle Green’s function. Calculating the self-energy operator in the approximation of noncrossing diagrams leads to a closed system of equations corresponding to the Migdal-Eliashberg strong-coupling theory. We propose a theory of high-temperature superconductivity due to kinematic interaction in the Hubbard model. We show that two pairing channels occur in systems with a strong Coulomb correlation: one due to the antiferromagnetic exchange in interband hopping and the other due to the coupling to spin and charge fluctuations in hopping within one Hubbard band. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 154, No. 1, pp. 129–146, January, 2008.     

Uniformity seminorms on ℓ<Superscript>∞</Superscript> and applications

A key tool in recent advances in understanding arithmetic progressions and other patterns in subsets of the integers is certain norms or seminorms. One example is the norms on ℤ/Nℤ introduced by Gowers in his proof of Szemerédi’s Theorem, used to detect uniformity of subsets of the integers. Another example is the seminorms on bounded functions in a measure preserving system (associated to the averages in Furstenberg’s proof of Szemerédi’s Theorem) defined by the authors. For each integer k ≥ 1, we define seminorms on ℓ^∞(ℤ) analogous to these norms and seminorms. We study the correlation of these norms with certain algebraically defined sequences, which arise from evaluating a continuous function on the homogeneous space of a nilpotent Lie group on a orbit (the nilsequences). Using these seminorms, we define a dual norm that acts as an upper bound for the correlation of a bounded sequence with a nilsequence. We also prove an inverse theorem for the seminorms, showing how a bounded sequence correlates with a nilsequence. As applications, we derive several ergodic theoretic results, including a nilsequence version of the Wiener-Wintner ergodic theorem, a nil version of a corollary to the spectral theorem, and a weighted multiple ergodic convergence theorem.     

Permittivity and one-particle distribution functions in the thermodynamics of a coulomb system

Using the grand canonical distribution and the virial theorem, we show that the Gibbs thermodynamic potential of a nonrelativistic system of charged particles is uniquely determined by its permittivity and the distribution functions of electrons and nuclei without using perturbation theory. This means that consistent approximations for the permittivity and one-particle distribution functions of electrons and nuclei must be used to calculate thermodynamic functions of the Coulomb system. To construct such selfconsistent approximations, we propose using a decoupling procedure based on separating the “connected” and “regular” parts of the temperature Green’s functions in the equations of motion. We consider the self-consistent Hartree-Fock approximation corresponding to this procedure.     

Multiplicity and vanishing lemmas for differential and <Emphasis Type="Italic">q</Emphasis>-difference equations in the Siegel–Shidlovsky theory

We present a general multiplicity estimate for linear forms in solutions of various types of functional equations, which extends the zero estimates used in some recent works on the Siegel–Shidlovsky theorem and its q-analogues. We also present a dual version of this estimate, as well as a new interpretation of Siegel’s theorem itself in terms of periods of Deligne’s irregular Hodge theory.     

Algebraic independence of the values ofE-functions at singular points and Siegel’s conjecture

We prove a generalization of Shidlovskii’s theorem on the algebraic independence of the values ofE-functions satisfying a system of linear differential equations that is well known in the theory of transcendental numbers. We consider the case in which the values ofE-functions are taken at singular points of these systems. Using the obtained results, we prove Siegel’s conjecture that, for the case of first-order differential equations, anyE-function satisfying a linear differential equation is representable as a polynomial in hypergeometricE-functions. Translated fromMatematicheskie Zametki, Vol. 67, No. 2, pp. 174–190, February, 2000.     

On the Equivalence and Generalized of Weyl Theorem Weyl Theorem

We know that an operator T acting on a Banach space satisfying generalized Weyl＇s theorem also satisfies Weyl＇s theorem. Conversely we show that if all isolated eigenvalues of T are poles of its resolvent and if T satisfies Weyl＇s theorem, then it also satisfies generalized Weyl＇s theorem. We give also a sinlilar result for the equivalence of a-Weyl＇s theorem and generalized a-Weyl＇s theorem. Using these results, we study the case of polaroid operators, and in particular paranormal operators.     

On estimates for integral solutions of linear inequalities

Recently, Bombieri and Vaaler obtained an interesting adelic formulation of the first and the second theorems of Minkowski in the Geometry of Numbers and derived an effective formulation of the well-known “Siegel’s lemma” on the size of integral solutions of linear equations. In a similar context involving linearinequalities, this paper is concerned with an analogue of a theorem of Khintchine on integral solutions for inequalities arising from systems of linear forms and also with an analogue of a Kronecker-type theorem with regard to euclidean frames of integral vectors. The proof of the former theorem invokes Bombieri-Vaaler’s adelic formulation of Minkowski’s theorem.     

Magnetic impurity effect on superconductivity in systems with comparable Fermi and Debye energies

We obtain the basic system of equations for a superconductivity theory of a system with a chaotically distributed paramagnetic substitution impurity in which the Migdal theorem is violated (the condition ω_D ≪E _F cannot be assumed). The electron-phonon and impurity diagrams and also the additional diagrams corresponding to intersections of the electron-phonon and electron-impurity lines are taken into account. In the weak electron-phonon coupling limit, we obtain an equation for the superconducting transition temperature T_C that differs from the corresponding equation for the usual superconductors by renormalizations of T_C0 and of the impurity scattering parameter, ρ. These quantities depend essentially on the Migdal parameter ω_D/E _F and on the transferred momentum q_c. We show that the decrease of T_C with the increase of the impurity concentration is slowed, as compared with usual superconductors, to an extent determined by m and q_c. We also evaluate the isotopic coefficient α, whose behavior as a function of the impurity concentration depends on m and q_c. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 119, No. 3, pp 455–474, June, 1999     

New diagram technique for the Anderson model

A generalized Wick theorem for the Anderson model is formulated. On this basis, the diagram technique for the temperature Green's function and the thermodynamic potential is developed. Particular attention is paid to the region of strong coupling of an impurity atom with a metal. The Dyson equations are solved and the impurity Green's function is expressed in terms of the mass operator. The contribution to the mass operator and corrections to the energies of localized states are obtained in the second order of perturbation theory.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 109, No. 2, pp. 279–294, November, 1996.     

Some remarks on Nagumo’s theorem

We provide a simpler proof for a recent generalization of Nagumo’s uniqueness theorem by A. Constantin: On Nagumo’s theorem. Proc. Japan Acad., Ser. A 86 (2010), 41–44, for the differential equation x′ = f(t, x), x(0) = 0 and we show that not only is the solution unique but the Picard successive approximations converge to the unique solution. The proof is based on an approach that was developed in Z. S. Athanassov: Uniqueness and convergence of successive approximations for ordinary differential equations. Math. Jap. 35 (1990), 351–367. Some classical existence and uniqueness results for initial-value problems for ordinary differential equations are particular cases of our result.     

Exponential sums,the geometry of hyperplane sections,and some diophantine problems

We estimate exponential sums with additive character along an affine variety given by a system of homogeneous equations, with a homogeneous function in the exponent. The proof uses the results of Deligne’s Weil Conjectures II and a generalization of Lefschetz hyperplane theorem to singular varieties. We apply our estimate to obtain an upperbound for the number of integer solutions of a system of homogeneous equations in a box. Another application is devoted to uniform distribution of solutions of a system of homogeneous congruences modulo a prime in the following sense: the portion of solutions in a box is proportional to the volume of the box, provided the box is not very small.     

Stability of Functional Equations in Several Variables

We prove a generalization of Hyers＇ theorem on the stability of approximately additive mapping and a generalization of Badora＇s theorem on an approximate ring homomorphism. We also obtain a more general stability theorem, which gives the stability theorems on Jordan and Lie homomorphisms. The proofs of the theorems given in this paper follow essentially the D. H. Hyers-Th. M. Rassias approach to the stability of functional equations connected with S. M. Ulam＇s problem.     

The 2×2 matrix Schlesinger system and the Belavin-Polyakov-Zamolodchikov system

We show that the Belavin-Polyakov-Zamolodchikov equation of the minimal model of conformal field theory with the central charge c = 1 for the Virasoro algebra is contained in a system of linear equations that generates the Schlesinger system with 2×2 tmatrices. This generalizes Suleimanov’s result on the Painlevé equations. We consider the properties of the solutions, which are expressible in terms of the Riemann theta function.     

Extremal dependence measure and extremogram: the regularly varying case

The dependence of large values in a stochastic process is an important topic in risk, insurance and finance. The idea of risk contagion is based on the idea of large value dependence. The Gaussian copula notoriously fails to capture this phenomenon. Two notions in a process or vector context which summarize extremal dependence in a function comparable to a correlation function are the extremal dependence measure (EDM) and the extremogram. We review these ideas and compare the two tools and end with a central limit theorem for a natural estimator of the EDM which allows drawing confidence bands comparable to those provided by Bartlett’s formula in a classical context of sample correlation functions.     

The diagram theory for the degenerate two-orbital hubbard model

We investigate the minimal model that takes orbital degrees of freedom into account: the degenerate two-orbital Hubbard model. Our consideration includes the intraatomic Coulomb interaction of two electrons with opposite spins on the same orbital and on different orbitals and interorbital hopping of tunneling electrons. We take the influence of states caused by Hund’s rule coupling on the metal-insulator phase transition into account. We generalize the diagram theory developed for strongly correlated orbitally nondegenerate systems to the case of orbital degeneration. For the one-particle renormalized Green’s function, we establish an equation of Dyson type for calculating the system spectral function using a simple approximation based on summing chain diagrams.     

Weyl’s Theorem and Perturbations

In the present paper we examine the stability of Weyl’s theorem under perturbations. We show that if T is an isoloid operator on a Banach space, that satisfies Weyl’s theorem, and F is a bounded operator that commutes with T and for which there exists a positive integer n such that Fⁿ is finite rank, then T + F obeys Weyl’s theorem. Further, we establish that if T is finite-isoloid, then Weyl’s theorem is transmitted from T to T + R, for every Riesz operator R commuting with T. Also, we consider an important class of operators that satisfy Weyl’s theorem, and we give a more general perturbation results for this class.